On functional inequalities associated with Drygas functional equation

TL;DR

This paper proves the equivalence between a specific functional inequality and the Drygas functional equation for functions mapping an abelian group to an inner product space, deepening understanding of their relationship.

Contribution

It establishes the equivalence between a particular functional inequality and the Drygas functional equation in the context of abelian groups and inner product spaces.

Findings

Equivalence of the inequality and the Drygas equation is proven.

The result applies to functions from abelian groups to inner product spaces.

The proof enhances understanding of functional equations and inequalities in this setting.

Abstract

In the paper, the equivalence of the functional inequality $$\|2f(x)+f(y)+f(-y)-f(x-y)\|\leq\|f(x+y)\|\;\;\;(x,y\in{G})$$ and the Drygas functional equation $$f(x+y)+f(x-y)=2f(x)+f(y)+f(-y)\;\;\;(x,y\in{G})$$ is proved for functions $f:G\rightarrow E$ where $(G, +)$ is an abelian group, $(E, <\cdot, \cdot>)$ is an inner product space, and the norm is derived from the inner product in the usual way.